Consistency, Truth and Ontology

Abstract

After a brief survey of the different meanings of consistency, the study is restricted to consistency understood as non-contradiction of sets of sentences. The philosophical reasons for this requirement are discussed, both in relation to the problem of sense and the problem of truth (also with historical references). The issue of mathematical truth is then addressed, and the different conceptions of it are put in relation with consistency. The formal treatment of consistency and truth in mathematical logic is then considered, with particular attention paid to the relation between syntactic and semantic properties of sets and calculi. After the crisis of mathematical intuition and the dominance of the formalistic view, it seemed that consistency could totally replace the requirement of truth in mathematics, also in the sense that the existence of “objects” of axiomatic systems could be granted by their consistency. A rejection of this claim is presented, whose central point is a detailed analysis of the theorem that any consistent set S of sentences of first order logic has a model. A critical scrutiny shows that this model is very peculiar, being offered by the elements of the same language that is being interpreted, and the satisfiability conditions for any sentence being constituted by the mere fact of belonging to S. Though not being insignificant from a metatheoretical point of view, this theorem fails to endow consistency (even in this privileged case) with an “ontological creativity”, that is, with the capability of providing a model ontologically distinct from the language itself (which is the precondition for the classical notion of truth that is also preserved in the Tarskian semantics and model theory). A final discussion regarding the different “ontological regions” and the referential nature of truth clarifies the different aspects of the whole issue discussed.